Problem: Simplify and expand the following expression: $ \dfrac{y}{5y - 4}+\dfrac{5y - 8}{5y - 2} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5y - 4)(5y - 2)$ Multiply the first term by $\dfrac{5y - 2}{5y - 2}$ $ \begin{align*} \dfrac{y}{5y - 4} \times \dfrac{5y - 2}{5y - 2} & = \dfrac{(y)(5y - 2)}{(5y - 4)(5y - 2)} \\ & = \dfrac{5y^2 - 2y}{(5y - 4)(5y - 2)}\end{align*} $ Multiply the second term by $\dfrac{5y - 4}{5y - 4}$ $ \begin{align*} \dfrac{5y - 8}{5y - 2} \times \dfrac{5y - 4}{5y - 4} & = \dfrac{(5y - 8)(5y - 4)}{(5y - 2)(5y - 4)} \\ & = \dfrac{25y^2 - 60y + 32}{(5y - 2)(5y - 4)}\end{align*} $ Now we have: $ = \dfrac{5y^2 - 2y}{(5y - 4)(5y - 2)} + \dfrac{25y^2 - 60y + 32}{(5y - 2)(5y - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5y^2 - 2y + 25y^2 - 60y + 32}{(5y - 4)(5y - 2)} $ $ = \dfrac{30y^2 - 62y + 32}{(5y - 4)(5y - 2)}$ Expand the denominator: $ = \dfrac{30y^2 - 62y + 32}{25y^2 - 30y + 8}$